John Bergdall (Boston University)
Overconvergent companion forms.

A theorem of Breuil and Emerton says that the p-adic Galois representation attached to a p-ordinary eigenform is split if and only if a certain overconvergent eigenform of negative weight exists. It is a characteristic zero analogue of the famous companion forms theorem of Gross modulo p. In this talk I will present a different proof of the result. The key point is that everything should be read off by studying the local geometry of the Coleman-Mazur eigencurve at a very special point. If time permits I will hint at generalizations and applications thereof.